Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem ∗ F. Mazenc, Projet MERE INRIA-INRA, UMR Analyse des Syst` emes et Biom´ etrie, INRA 2, pl. Viala, 34060 Montpellier, France, email: Frederic.Mazenc@ensam.inra.fr D. Neˇ si´ c, Dept. of Electrical and Electronic Engineering, The University of Melbourne, Vic 3052, Australia, email: d.nesic@ee.mu.oz.au, May 19, 2006 Abstract The classical Matrosov theorem concludes uniform asymptotic stability of time varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi- definite derivative along solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero [M1]. Recently, sev- eral generalizations of the classical Matrosov theorem that use a finite number of Lyapunov- like functions have been reported in [LPPT2]. None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite deriva- tive along solutions) that is a very useful analysis and controller design tool for nonlinear systems. We provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller. Key words. Lyapunov functions, Matrosov Theorem, Nonlinear, Stability, Time-Varying. 1 Introduction Lyapunov second method is ubiquitous in stability and robustness analysis of nonlinear systems. In recent years, its different versions were used for controller design, e.g. control Lyapunov functions, nonlinear damping, backstepping, forwarding, and so on [K, SJK, ST, FP, M2]. While it is often useful to obtain a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) to analyze robustness or redesign the given controller, it is often the case that only a weak Lyapunov function (positive definite, decrescent, ∗ This work was supported by the Australian Research Council under the Discovery Grants Scheme while the first author was visiting The University of Melbourne. 1